We performed a theoretical investigation of the progression of a finger of air through a liquid-filled flexible-walled channel - an initial model of pulmonary airway reopening. Positive pressure, Pb* drives the bubble forward, and separates flexible walls that are modelled as membranes under tension, T, supported by linearly elastic springs with elasticity K. The gap width between the walls under stress-free conditions is 2H, and the liquid has constant surface tension, γ, and viscosity, μ. Three parameters define the state of the system: Ca = μU/γ is a dimensionless velocity that represents the ratio of viscous to capillary stresses; η = T/γ is the wall tension to surface tension ratio, and γ = KH2/γ is the wall elastance parameter. We examined steady-state solutions as a function of these parameters using lubrication analysis and the boundary element method.
These studies showed multiple-branch behaviour in the Pb-Ca relationship, where Pb = Pb*/(γ/H) is the dimensionless bubble pressure. Low Ca flows (Ca [Lt ] min (1, (Γ3/η)1/2)) are dominated by the coupling of surface tension and elastic stresses. In this regime, Pb decreases as Ca increases owing to a reduction in the downstream resistance to flow, caused by the shortening of the section connecting the open end of the channel to the fully collapsed region. High Ca behaviour (max (1, (γ3/η)1/2) [Lt ] Ca [Lt ] η) is dominated by the balance between fluid viscous and longitudinal wall tension forces, resulting in a monotonically increasing Pb–Ca relationship. Increasing η or decreasing Γ reduces the Ca associated with the transition from one branch to the other. Low Ca streamlines show closed vortices at the bubble tip, which disappear with increasing Ca.
Start-up yield pressures are predicted to range from 1 [les ] Pyield*/(γ/L*) [les ] 2, which is less than the minimum pressure for steady-state reopening, Pmin/(γ/L*), where L* is the upstream channel width. Since Pyield* < Pmin*, the theory implies that low Ca reopening may be unsteady, a behaviour that has been observed experimentally. Our results are consistent with experimental observations showing that Pb* in highly compliant channels scales with γ/L*. In contrast, we find that wall shear stress scales with γ/H. These results imply that wall shear and normal stresses during reopening are potentially very large and may be physiologically significant.